Even Faster Ideal Point Estimation
Using Variational Autoencoders for Item Response Theory
MIT
December 1, 2025
Project Overview
Researchers with discrete data (parliamentary votes, campaign websites, survey responses) often want to estimate latent traits (ideology, policy positions, personality)
One common choice is Item Response Theory (IRT), a parametric approach
- Estimation of IRT models is computationally expensive; speedups exist but are still infeasible for very large datasets (think: millions of observations) or complex models (e.g., multiple latent dimensions)
- I introduce Variational Autoencoders (VAEs) for estimating IRT models to political science, a connection previously developed in psychology and computer science
Item-Response Theory
Basics
The canonical “2-parameter” IRT model: \[\pi_{ij} = Pr(y_{ij} = c | \alpha_i, \gamma_{j}, \beta_{j}) = F(\underbrace{\alpha_i}_{\text{Ideal Point}} \cdot \underbrace{\gamma_{j}}_{\text{Discrimination}} - \underbrace{\beta_{j}}_{\text{Difficulty}}),\] where \(F(\cdot)\) is a monotone mapping function (e.g., inverse logit or softmax), \(i\) indexes respondents and \(j\) indexes questions. Multiple dimensions are also supported.
Construct a likelihood: \(\mathcal{L} = \prod_{i=1}^N \prod_{j=1}^J \pi_{ij}^{y_{ij}}(1-\pi_{ij}^{1-y_{ij}})\), a prior for the latent traits \(f(\alpha_i) \sim \mathcal{N}(0, 1),\) and solve for the posterior: \(f(\alpha | y) \propto \mathcal{L_i} f(\alpha_i)\)
Item-Response Theory
Estimation Approaches
Current standards in political science:
- Bayesian MCMC (Jackman 2009)
- EM Algorithm: Closed form for certain models (Imai, Lo, and Olmsted 2016; Peress 2022)
- Variational Inference: Instead of directly optimizing the marginal log-likelihood, estimate a simpler function \(\hat g(\cdot)\) for each posterior \(f(\alpha | y)\) (Grimmer 2011)
My proposal from psychometrics and computer science:
- Variational Autoencoder: Instead of estimating a separate \(\hat g(\cdot)\) for each individual, estimate a single, flexible model that predicts the posterior distribution for an individual given their responses (Curi et al. 2019)
Variational Autoencoders
Some Details
Instead of estimating a separate \(\hat g(\cdot)\) for each individual, estimate a single, flexible model that predicts the posterior distribution for an individual given their responses.
We model latent traits \(\alpha_i \sim \text{MVN}(\mu_i, \sigma_i)\) where \((\mu_i, \sigma_i) = \text{NeuralNet}_\mathbf{\phi} (\mathbf{y}_i)\)
Combining them together yields a Variational Autoencoder (Kingma and Welling 2014), a kind of structured neural network model. Parameters are estimated by:
\[\begin{align*} \left( \hat{\mathbf{\phi}}, \hat \gamma, \hat \beta \right) &= \underset{\mathbf{\phi}, \gamma, \beta}{argmax} \left( \mathbb{E}\left[ \log(p(\mathbf{y}_i | \hat \alpha_i ; \gamma_j ; \beta_j)) \right] - \text{KL}\left[ g(\alpha_i | \mathbf{y}_i; \mathbf{\phi}) ||\ f(\alpha_i | \mathbf{y}_i;) \right] \right) \\ &= argmax \text{Evidence Lower Bound (ELBO)} \end{align*}\]
Variational Autoencoders
Model Diagram
Variational Autoencoders
Missing Data
Missing data is common in IRT applications (e.g., legislators don’t vote on every bill, survey respondents skip questions, voters don’t cast a ballot in every contest)
- VAEs and machine learning models in general don’t handle missing data naturally
- Passing a “mask” explicitly to the model is better than dropping, imputing along the way, or a mix of approaches (Veldkamp, Grasman, and Molenaar 2025)
Variational Autoencoders
Computation
Bespoke model built in PyTorch, then fit for N epochs on MIT’s Engaging Cluster until satisfactory convergence was reached
- About 1h30m for 1m voters x 5000 contests x 4 dimensions
- Probably a couple months as Bayesian MCMC, a couple weeks with Variational Inference, and infeasible with EM
Example Usage
Candidate-Agnostic Ideological Scores
Example Usage
Uncontested District Attorney Contests in Colorado
Example Usage
Ballot Questions in Colorado
Thank you!
Mason Reece | mpreece@mit.edu
VAEs for IRT